cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A360612 Number of binary operators defined on the finite chain L_n={0,1,...n}, C:L_n^2-> L_n, which are increasing in each argument, and satisfy the boundary conditions C(0,n)=C(n,0)=0 and C(n,n)=n.

Original entry on oeis.org

1, 14, 805, 208152, 250409016, 1423422089804, 38533696399916432, 4988815527667401921920, 3096067500138473517778378240, 9222307552079662925642825622240000, 131945758198070262889738914466064452265625, 9070830675953705403006049148134626173379375000000
Offset: 1

Views

Author

Marc Munar, Feb 14 2023

Keywords

Links

Crossrefs

Cf. A008793, A071095.

Programs

  • Mathematica
    Table[Product[Product[Product[(i + j + k - 1)/(i + j + k - 2), {k, 1, n}], {j, 1, n}], {i, 1, n}] - Product[Product[Product[(i + j + k - 1)/(i + j + k - 2), {k, 1, n - 1}], {j, 1, n}], {i, 1, n}], {n, 1, 15}]

Formula

a(n) = Product_{i=1..n} Product_{j=1..n} Product_{k=1..n} (i+j+k-1)/(i+j+k-2) - Product_{i=1..n} Product_{j=1..n} Product_{k=1..n-1} (i+j+k-1)/(i+j+k-2).
a(n) = A008793(n+1) - A071095(n). - Vaclav Kotesovec, Nov 18 2023

A071094 Number of ways to tile hexagon of edges n, n, n+1, n, n, n+1 with diamonds of side 1.

Original entry on oeis.org

1, 3, 50, 4116, 1646568, 3184461423, 29706808370096, 1335119245893326400, 288882990167192721013376, 300792059519113653077154558000, 1506680146887473588202049621593937500, 36298820709557430183399305000196605531250000, 4205446372314569673006362329181090368935937500000000, 2342761095072644391194625697884219372917666852341417500000000
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2002

Keywords

References

  • J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see page 261).

Links

Crossrefs

Cf. A008793, A071095, A103905.

Programs

  • Mathematica
    Table[Product[(i+j+k+2)/(i+j+k+1),{i,0,n-1},{j,0,n-1},{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Apr 26 2015 *)
  • PARI
    a(n) = prod(k=0, n, binomial(2*n+k,n+k)/binomial(n+k,k)) \\ Michel Marcus, May 20 2013

Formula

a(n) = Product_{i=0..a-1} Product_{j=0..b-1} Product_{k=0..c-1} (i+j+k+2)/(i+j+k+1) with a=b=n, c=n+1.
a(n) = Product_{k=0..n} C(2n+k,n+k)/C(n+k,k). - Paul Barry, May 13 2008
a(n) ~ exp(1/12) * 3^(9*n^2/2 + 3*n + 5/12) / (A * n^(1/12) * 2^(6*n^2 + 4*n + 3/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 26 2015

A071096 Number of ways to tile hexagon of edges n, n+1, n+2, n, n+1, n+2 with diamonds of side 1.

Original entry on oeis.org

1, 10, 490, 116424, 133613766, 739309710568, 19702998159210080, 2527580342020127455360, 1560172391098377453031770400, 4632518859090968506120863642225000, 66153724447703043353053979949899667187500, 4542776083800437392420665771479758969781250000000, 1499928882906010042230116408158354282455601808812500000000
Offset: 0

Views

Author

N. J. A. Sloane, May 28 2002

Keywords

References

  • J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see page 261).

Links

Crossrefs

Cf. A008793, A071094, A071095.

Programs

  • Mathematica
    Table[Product[Product[Product[(i+j+k-1)/(i+j+k-2),{i,1,n+1}],{j,1,n+1}],{k,1,n+1}],{n,0,10}]/2 (* Alexander Adamchuk, Jul 10 2006 *)
  • PARI
    {a(n) = abs(matdet(matrix(n+1, n+1, i, j, binomial(2*(n+1), i+j))))}; \\ Shifted by Georg Fischer, Jun 19 2022

Formula

Product_{i=0..a-1} Product_{j=0..b-1} Product_{k=0..c-1} (i+j+k+2)/(i+j+k+1) with a=n, b=n+1, c=n+2.
a(n) = (-1)^floor((n+1)/2)*det(M(n+1)) where M(n) is the n X n matrix m(i, j)=C(2n, i+j), with i and j ranging from 1 to n. - Benoit Cloitre, Jan 31 2003
a(n) = (1/2)*Product[Product[Product[(i+j+k-1)/(i+j+k-2),{i,1,n+1}],{j,1,n+1}],{k,1,n+1}]. a(n) = A008793(n+1)/2. - Alexander Adamchuk, Jul 10 2006
a(n) ~ exp(1/12) * 3^(9*n^2/2 + 9*n + 53/12) / (A * n^(1/12) * 2^(6*n^2 + 12*n + 27/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Apr 26 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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